The combination of an inductor and a capacitor in a loop, typically known as a resonant circuit, is an important building block in electronic circuits. Oscillators, filters and impedances at the input or output of radio frequency (RF) amplifiers, for example, exploit the highly frequency-selective nature of resonant circuits to create, enhance or eliminate signals at particular frequencies. FIG. 1a shows a well-known resonant circuit 1, which comprises a single inductor 11 and a single capacitor 12 in a loop. The resonant frequency, at which the total impedance around the loop is zero, is inversely proportional to the square-root of the product between inductance L and capacitance C. In many applications, for example tunable filters and voltage-controlled oscillators, it is desirable to vary the resonant frequency. This can be achieved by varying either the capacitance 12 or the inductance 11, as shown in FIG. 1b and FIG. 1c, respectively. Although various mechanical means to vary or control the value of a capacitor or inductor have existed for some time, electronically variable capacitors, or varactors, are the most frequently used today in miniaturized systems such as an RF integrated circuit containing a voltage-controlled oscillator (VCO).
In electro-mechanical resonators based on a piezoelectric ceramic resonator or quartz crystal, a variable capacitor, found in oscillators such as that shown in FIG. 2a, is also sometimes used; the electrical equivalent circuit of the crystal is shown in FIG. 2b. The equivalent circuit comprises an inductance 21 and a capacitance 22 connected in series between the terminals of device in parallel with a second capacitance 23.
Although the equivalent capacitance in the resonator is the series combination of motional capacitance Cs 22 and case capacitance Cp 23, only the case capacitance is accessible from the crystal's two terminals 25 and 25′, therefore its resonant frequency is controllable only by adding to it a variable capacitance in parallel, as shown in FIG. 2c. Control of the resonant frequency is desirable, to correct any inaccuracy in it due for example to manufacturing tolerances, temperature dependence and aging (In FIG. 2c CP represents the case capacitance and the variable capacitance, which are connected in parallel). A circuit containing such a crystal is usually called a voltage-controlled crystal oscillator (VCXO) and its frequency is set by using a varactor for the variable capacitor and varying the bias voltage varies its capacitance.
The motional capacitance Cs in such resonators is typically several orders of magnitude smaller than the case capacitance Cp, so that increasing Cp has limited effectiveness in decreasing the resonant frequency fp, as can be seen from equation (1). Since the fixed part
                                                                        f                p                            =                              1                                  2                  ⁢                  π                  ⁢                                                                                    L                        s                                            ⁢                                                                                                    C                            s                                                    ⁢                                                      C                            P                                                                                                                                C                            s                                                    +                                                      C                            P                                                                                                                                                                                                                      =                                                1                                      2                    ⁢                    π                    ⁢                                                                                            L                          s                                                ⁢                                                  C                          s                                                                                                                    ⁢                                                      1                    +                                                                  C                        s                                                                    C                        P                                                                                                                                                    (        1        )            of Cp due to the casing of a ceramic or crystal resonator is usually already quite large, adding further variable capacitance to Cp results in diminishing return in terms of relative frequency shift. (The right hand square root simply tends closer to 1). The tuning range of a voltage-controlled crystal oscillator (VCXO) is therefore typically limited.